## Global continuum and multiple positive solutions to a $$p$$-Laplacian boundary-value problem.(English)Zbl 1260.34045

Summary: A $$p$$-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from $$(\lambda,u)=(0,0)$$ is shown, and it can be extended to $$\lambda=\infty$$. Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any $$\lambda>0$$; thus the continuum $$\mathcal C$$ is indeed a continuous curve globally defined for all $$\lambda>0$$. In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
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